Project Graph Energy and the Completion Number Program Purdue Experimental Mathematics Lab Term Spring 2026 Status Accepted Research Area Mathematics Description Graph energy is a linear-algebraic invariant of graphs introduced in the 1970s from theoretical chemistry and can be described as the sum of the absolute values of the eigenvalues of the adjacency matrix of the graph. The completion number of a graph was more recently introduced and gives an integer value which quantifies the defect in being able to complete a matrix which is partially positive on the edges of the graph to a positive matrix. The project will focus on finding effective, computable bounds for the completion number of a graph and to exploring connections between the completion number of a graph, graph energy, and other invariants. Time permitting we will also seek to explore how graph energy and the completion number can be adapted to the setting of "quantum graphs" - subspaces of square matrices closed under transpose and containing the identity. Supervisor Thomas J Sinclair Mentor Type of work required Course-based, vertically-integrated research projects in mathematics. Each project will consist of a small research team consisting of typically 2-4 undergraduates, a graduate mentor, and a faculty mentor. The graduate mentor and undergraduates will meet on a weekly basis, with full team meetings every few weeks as determined by the faculty mentor. To apply include a brief (one page or less) statement explaining your interest in mathematics research. Additionally, list all mathematics courses you have taken with your grade in each one, as well as any other coursework or qualifications that you feel are pertinent. Undergraduates who have been accepted into a project must sign up for the 3-credit "Purdue Experimental Math Lab" course (currently listed under MA 490) and must pledge that they are able to dedicate 10 hours of effort per week to the project. Websites https://www.math.purdue.edu/pxml/join-pxml.html Qualifications Completion of a proof-based course in Linear Algebra. Some experience with Python is preferred. Ability to meet Wednesday afternoon or evening Credit Hours 3 Weekly Hours 10 (estimated)